高维数据几何结构及降维

高维数据几何结构及降维

王建忠, 编著

出版社:高等教育出版社

年代:2011

定价:45.6

书籍简介:

本书以数据的几何结构为框架介绍了各种不同的数据降维方法以及它们的数学原理和计算程序,除了介绍经典的线性降维方法, 如主分量分析和线性多维度量法外, 本书更着重于非线性方法的探讨, 尤其对近期发展起来的非线性方法,如几何扩散映射法, 极大方差展开法, Hessian局部线性嵌入, 以及局部切平面校准等方法作出了深入细致的讨论, 并给出了这些方法与传统的非线性方法如局部线性嵌入和等距映射法的比较. 对每个方法的主要思想, 数学依据, 算法和程序都作了详细描述. 对于大规模数据的降维计算, 本书则着重介绍了界标抽样法和随机化算法以克服计算机计算过程中遇到的内存不足和计算不稳定的困难。

作者介绍:

王建忠,1967年毕业于北京大学数学力学系,1981年获浙江大学应用数学系硕士学位,后执教于武汉大学并于1988年晋升为教授,现任美国得克萨斯州山姆休斯顿大学教授。主要研究方向包括:样条和逼近理论、小波分析、图像处理以及高维数据分析等。曾主持多项美国自然科学基金及其他基金课题的研究。发表学术论文70余篇,专利1项。

书籍目录:

Chapter 1 Introduction

1.1 0verview of Dimensionality R,eduction

1.2 High Dimension Data Acquisition

1.2.1 Collection of Images in Face Recognition

1.2.2 Handwriting Letters and Digits

1.2.3 Text Documents

1.2.4 Hyperspectral Images

1.3 Curse of the Dimensionality

1.3.1 Volume of Cubes and Spheres

1.3.2 Volume of a Thin Spherical Shell

1.3.3 Tail Probability of the Multivariate Gaussian Distributions

1.3.4 Diagonals of Cube

1.3.5 Concentration of Norms and Distances

1.4 Intrinsic and Extrinsic Dimensions

1.4.1 Intrinsic Dimension Estimation

1.4.2 Correlation Dimension

1.4.3 Capacity Dimension

1.4.4 Multiscale Estimation

1.5 0utline of the Book

1.5.1 Categories of DR Problems

1.5.2 Scope of This Book

1.5.3 0ther Topics Related to This Book

1.5.4 Artificial Surfaces for Testing DR Algorithms

Part I Data Geometry

Chapter 2 Preliminary Calculus on Manifolds

2.1 Linear Manifold

2.1.1 Subspace and Projection

2.1.2 Functions on Euclidean Spaces

2.1.3 Laplace Operator and Heat Diffusion Kernel

2.2 Differentiable Manifolds

2.2.1 Coordinate Systems and Parameterization

2.2.2 Tangent Spaces and Tangent Vectors

2.2.3 Riemannian Metrics

2.2.4 Geodesic Distance

2.3 Functions and Operators on Manifolds

2.3.1 Functions on Manifolds

2.3.2 0perators on Manifolds

Chapter 3 Geometric Structure of High-Dirnensional

3.1 Similarity and Dissimilarity of Data

3.1.1 Neighborhood Definition

3.1.2 Algorithms for Construction of Neighborhood

3.2 Graphs on Data Sets

3.2.1 Undirected Graphs

3.2.2 Directed Graphs

3.2.3 Neighborhood and Data Graphs

3.3 Spectral Analysis of Graphs

3.3.1 Laplacian of Graphs

3.3.2 Laplacian on Weighted Graphs

3.3.3 Contracting Operator on Weighted Graph

Chapter 4 Data Models and Structures of Kernels of DR

4.1 Data Models in Dimensionality Reduction

4.1.1 Input Data of First Type

4.1.2 Input Data of Second Type

4.1.3 Constraints on Output Data

4.1.4 Consistence of Data Graph

4.1.5 Robust Graph Connection Algorithm

4.2 Constructions of DR Kernels

4.2.1 DR Kernels of Linear Methods

……

Part Ⅱ Linear Dimensionality reduction

Part Ⅲ Nonlinear Dimensionality Reduction

内容摘要:

Many objects in our world can be electronically represented with high-dimensional data- speech signals, images, videos, electrical text documents.We often need to analyze a large amount of data and process them. However,due to the high dimension of these data, directly processing them using reg-ular systems may be too complicated and unstable to be feasible. In order toprocess high-dimensional data, dimensionality reduction technique becomescrucial. Dimensionality reduction is a method to represent high-dimensionaldata by their low-dimensional embeddings so that the low-dimensional data can be effectively used either in processing systems, or for better understand-ing. This technique has proved an important tool and has been widely used in many fields of data analysis, data mining, data visualization, and machine learning.

书籍规格:

书籍详细信息
书名高维数据几何结构及降维站内查询相似图书
9787040317046
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出版地北京出版单位高等教育出版社
版次1版印次1
定价(元)45.6语种英文
尺寸24 × 16装帧精装
页数印数 270

书籍信息归属:

高维数据几何结构及降维是高等教育出版社于2011.10出版的中图分类号为 O212.1 的主题关于 统计数据-统计分析(数学)-英文 的书籍。